The proposed research will develop and test statistical semiparametric methods for estimation of a (multivariate) distribution of a (multivariate) time-random variable of interest when the observed time is censored (data which is inexact) and when censored/uncensored, possibly high dimensional, time-dependent and/or time-independent covariates are available. The developed methods have two imporant extensions to the existing methods: (1) they will allow for dependence of the time random variable of interest and the censoring random variables; (2) they are able to use additional time-dependent and high-dimensional time-independent covariates in a nonparametric and locally optimal way. For example, in the study of infectious diseases which change the immune system, a large gain in efficiency of the estimation method can be obtained by modelling the relationship between responses of the immune system and the time-variable of interest. The estimators resulting from the developed methods are consistent under few modelling assumptions, but their efficiency will depend on how well one is able to model the relationships between time of interest and the time-dependent and/or time-independent covariates. The method will be applied to data from animal carcinogenicity experiments and other cancer studies for estimation of the distribution of the time of onset of tumors, to AIDS data (infectious disease data) and the method will also be used for estimation of the survival function of an implant (liver shunt) in the human body. The data available contain, beyond a large number of covariates, time-dependent covariates with predictable value for the time variable of interest. The performance of the method will be studied by applying it to simulated data where the true models are known. Simulations will also be used to do an efficiency comparison of the methods with existing methods.